Shear sum rule in higher derivative gravity theories

We study holographic shear sum rules in Einstein gravity with curvature squared corrections. Sum rules relate weighted integral over spectral densities of retarded correlators in the shear channel to the one point functions of the CFTs. The proportionality constant can be written in terms of the data of three point functions of the stress tenors of the CFT ($t_2$ and $t_4$). For CFTs dual to two derivative Einstein gravity, this proportionality constant is just $\frac{d}{2(d+1)}$. This has been verified by a direct holographic computation of the retarded correlator for Einstein gravity in $AdS_{d+1}$ black hole background. We compute corrections to the holographic shear sum rule in presence of higher derivative corrections to the Einstein-Hilbert action. We find agreement between the sum rule obtained from a general CFT analysis and holographic computation for Gauss Bonnet theories in $AdS_5$ black hole background. We then generalize the sum rule for arbitrary curvature squared corrections to Einstein-Hilbert action in $d\geq 4$. Evaluating the parameters $t_2$ and $t_4$ for the possible dual CFT in presence of such curvature corrections, we find an agreement with the general field theory derivation to leading order in coupling constants of the higher derivative terms.